墨卡托投影(Mercator Projection),又名“等角正轴圆柱投影”,荷兰地图学家墨卡托(Mercator)在1569年拟定,假设地球被围在一个中空的圆柱里,其赤道与圆柱相接触,然后再假想地球中心有一盏灯,把球面上的图形投影到圆柱体上,再把圆柱体展开,这就是一幅标准纬线为零度(即赤道)的“墨卡托投影”绘制出的世界地图。
一、墨卡托投影坐标系(Mercator Projection)
墨卡托投影以整个世界范围,赤道作为标准纬线,本初子午线作为中央经线,两者交点为坐标原点,向东向北为正,向西向南为负。南北极在地图的正下、上方,而东西方向处于地图的正右、左。
由于Mercator Projection在两极附近是趋于无限值得,因此它并没完整展现了整个世界,地图上最高纬度是85.05度。为了简化计算,我们采用球形映射,而不是椭球体形状。虽然采用Mercator Projection只是为了方便展示地图,需要知道的是,这种映射会给Y轴方向带来0.33%的误差。
由于赤道半径为6378137米,则赤道周长为2*PI*r = 20037508.3427892,因此X轴的取值范围:[-20037508.3427892,20037508.3427892]。当纬度φ接近两极,即90°时,Y值趋向于无穷。因此通常把Y轴的取值范围也限定在[-20037508.3427892,20037508.3427892]之间。因此在墨卡托投影坐标系(米)下的坐标范围是:最小为(-20037508.3427892, -20037508.3427892 )到最大 坐标为(20037508.3427892, 20037508.3427892)。
二、地理坐标系(Geographical coordinates)
地理经度的取值范围是[-180,180],纬度不可能到达90°,通过纬度取值范围为[20037508.3427892,20037508.3427892]反计算可得到纬度值为85.05112877980659。因此纬度取值范围是[-85.05112877980659,85.05112877980659]。因此,地理坐标系(经纬度)对应的范围是:最小地理坐标(-180,-85.05112877980659),最大地理坐标(180, 85.05112877980659)。
三、地面分辨率(Ground Resolution)
地面分辨率是以一个像素(pixel)代表的地面尺寸(米)。以微软Bing Maps为例,当Level为1时,图片大小为512*512(4个Tile),那么赤道空间分辨率为:赤道周长/512。其他纬度的空间分辨率则为 纬度圈长度/512,极端的北极则为0。Level为2时,赤道的空间分辨率为 赤道周长/1024,其他纬度为 纬度圈长度1024。很明显,Ground Resolution取决于两个参数,缩放级别Level和纬度latitude ,Level决定像素的多少,latitude决定地面距离的长短。
地面分辨率的公式为,单位:米/像素:
ground resolution = (cos(latitude * pi/180) * 2 * pi * 6378137 meters) / (256 * 2level pixels)
最低地图放大级别(1级),地图是512 x 512像素。每下一个放大级别,地图的高度和宽度分别乘于2:2级是1024 x 1024像素,3级是2048 x 2048像素,4级是4096 x 4096像素,等等。通常而言,地图的宽度和高度可以由以下式子计算得到:map width = map height = 256 * 2^level pixels
四、地图比例尺(Map Scale)
地图比例尺是指测量相同目标时,地图上距离与实际距离的比例。通过地图分辨率在计算可知由Level可得到图片的像素大小,那么需要把其转换为以米为单位的距离,涉及到DPI(dot per inch),暂时可理解为类似的PPI(pixelper inch),即每英寸代表多少个像素。256 * 2level / DPI 即得到相应的英寸inch,再把英寸inch除以0.0254转换为米。实地距离仍旧是:cos(latitude * pi/180) * 2 * pi * 6378137 meters; 因此比例尺的公式为:
map scale = 256 * 2level / screen dpi / 0.0254 / (cos(latitude * pi/180) * 2 * pi * 6378137)
比例尺= 1 : (cos(latitude * pi/180) * 2 * pi * 6378137 * screen dpi) / (256 * 2level * 0.0254)
地面分辨率和地图比例尺之间的关系:
map scale = 1 : ground resolution * screen dpi / 0.0254 meters/inch
缩放级别 |
地图宽度、高度(像素) |
地面分辨率(米/像素) |
地图比例尺(以96dpi为例) |
1 |
512 |
78,271.5170 |
1 : 295,829,355.45 |
2 |
1,024 |
39,135.7585 |
1 : 147,914,677.73 |
3 |
2,048 |
19,567.8792 |
1 : 73,957,338.86 |
4 |
4,096 |
9,783.9396 |
1 : 36,978,669.43 |
5 |
8,192 |
4,891.9698 |
1 : 18,489,334.72 |
6 |
16,384 |
2,445.9849 |
1 : 9,244,667.36 |
7 |
32,768 |
1,222.9925 |
1 : 4,622,333.68 |
8 |
65,536 |
611.4962 |
1 : 2,311,166.84 |
9 |
131,072 |
305.7481 |
1 : 1,155,583.42 |
10 |
262,144 |
152.8741 |
1 : 577,791.71 |
11 |
524,288 |
76.4370 |
1 : 288,895.85 |
12 |
1,048,576 |
38.2185 |
1 : 144,447.93 |
13 |
2,097,152 |
19.1093 |
1 : 72,223.96 |
14 |
4,194,304 |
9.5546 |
1 : 36,111.98 |
15 |
8,388,608 |
4.7773 |
1 : 18,055.99 |
16 |
16,777,216 |
2.3887 |
1 : 9,028.00 |
17 |
33,554,432 |
1.1943 |
1 : 4,514.00 |
18 |
67,108,864 |
0.5972 |
1 : 2,257.00 |
19 |
134,217,728 |
0.2986 |
1 : 1,128.50 |
20 |
268,435,456 |
0.1493 |
1 : 564.25 |
21 |
536,870,912 |
0.0746 |
1 : 282.12 |
22 |
1,073,741,824 |
0.0373 |
1 : 141.06 |
23 |
2,147,483,648 |
0.0187 |
1 : 70.53 |
五、Bing Maps像素坐标系和地图图片编码
为了优化地图系统性能,提高地图下载和显示速度,所有地图都被分割成256 x 256像素大小的正方形小块。由于在每个放大级别下的像素数量都不一样,因此地图图片(Tile)的数量也不一样。每个tile都有一个XY坐标值,从左上角的(0, 0)至右下角的(2^level–1, 2^level–1)。例如在3级放大级别下,所有tile的坐标值范围为(0, 0)至(7, 7),如下图:
![GIS理论(墨卡托投影、地理坐标系、地面分辨率、地图比例尺、Bing Maps Tile System)](http://img.e-com-net.com/image/product/8d7002d167e843f5913f857c90830759.jpg)
已知一个像素的XY坐标值时,我们很容易得到这个像素所在的Tile的XY坐标值:
tileX = floor(pixelX / 256)
tileY = floor(pixelY / 256)
为了简化索引和存储地图图片,每个tile的二维XY值被转换成一维字串,即四叉树键值(quardtree key,简称quadkey)。每个quadkey独立对应某个放大级别下的一个tile,并且它可以被用作数据库中B-tree索引值。为了将坐标值转换成quadkey,需要将Y和X坐标二进制值交错组合,并转换成4进制值及对应的字符串。例如,假设在放大级别为3时,tile的XY坐标值为(3,5),quadkey计算如下:
tileX = 3 = 011(二进制)
tileY = 5 = 101(二进制)
quadkey = 100111(二进制)
= 213(四进制)
= “213”
Quadkey还有其他一些有意思的特性。第一,quadkey的长度等于该tile所对应的放大级别;第二,每个tile的quadkey的前几位和其父tile(上一放大级别所对应的tile)的quadkey相同,下图中,tile 2是tile 20至23的父tile,tile 13是tile 130至133的父级:
最后,quadkey提供的一维索引值通常显示了两个tile在XY坐标系中的相似性。换句话说,两个相邻的tile对应的quadkey非常接近。这对于优化数据库的性能非常重要,因为相邻的tile通常被同时请求显示,因此可以将这些tile存放在相同的磁盘区域中,以减少磁盘的读取次数。
下面是微软Bing Maps的TileSystem相关算法:
using
System;
using
System.Text;
namespace
Microsoft.MapPoint
{
static
class
TileSystem
{
private
const
double
EarthRadius
=
6378137
;
private
const
double
MinLatitude
=
-
85.05112878
;
private
const
double
MaxLatitude
=
85.05112878
;
private
const
double
MinLongitude
=
-
180
;
private
const
double
MaxLongitude
=
180
;
///
<summary>
///
Clips a number to the specified minimum and maximum values.
///
</summary>
///
<param name="n">
The number to clip.
</param>
///
<param name="minValue">
Minimum allowable value.
</param>
///
<param name="maxValue">
Maximum allowable value.
</param>
///
<returns>
The clipped value.
</returns>
private
static
double
Clip(
double
n,
double
minValue,
double
maxValue)
{
return
Math.Min(Math.Max(n, minValue), maxValue);
}
///
<summary>
///
Determines the map width and height (in pixels) at a specified level
///
of detail.
///
</summary>
///
<param name="levelOfDetail">
Level of detail, from 1 (lowest detail)
///
to 23 (highest detail).
</param>
///
<returns>
The map width and height in pixels.
</returns>
public
static
uint
MapSize(
int
levelOfDetail)
{
return
(
uint
)
256
<<
levelOfDetail;
}
///
<summary>
///
Determines the ground resolution (in meters per pixel) at a specified
///
latitude and level of detail.
///
</summary>
///
<param name="latitude">
Latitude (in degrees) at which to measure the
///
ground resolution.
</param>
///
<param name="levelOfDetail">
Level of detail, from 1 (lowest detail)
///
to 23 (highest detail).
</param>
///
<returns>
The ground resolution, in meters per pixel.
</returns>
public
static
double
GroundResolution(
double
latitude,
int
levelOfDetail)
{
latitude
=
Clip(latitude, MinLatitude, MaxLatitude);
return
Math.Cos(latitude
*
Math.PI
/
180
)
*
2
*
Math.PI
*
EarthRadius
/
MapSize(levelOfDetail);
}
///
<summary>
///
Determines the map scale at a specified latitude, level of detail,
///
and screen resolution.
///
</summary>
///
<param name="latitude">
Latitude (in degrees) at which to measure the
///
map scale.
</param>
///
<param name="levelOfDetail">
Level of detail, from 1 (lowest detail)
///
to 23 (highest detail).
</param>
///
<param name="screenDpi">
Resolution of the screen, in dots per inch.
</param>
///
<returns>
The map scale, expressed as the denominator N of the ratio 1 : N.
</returns>
public
static
double
MapScale(
double
latitude,
int
levelOfDetail,
int
screenDpi)
{
return
GroundResolution(latitude, levelOfDetail)
*
screenDpi
/
0.0254
;
}
///
<summary>
///
Converts a point from latitude/longitude WGS-84 coordinates (in degrees)
///
into pixel XY coordinates at a specified level of detail.
///
</summary>
///
<param name="latitude">
Latitude of the point, in degrees.
</param>
///
<param name="longitude">
Longitude of the point, in degrees.
</param>
///
<param name="levelOfDetail">
Level of detail, from 1 (lowest detail)
///
to 23 (highest detail).
</param>
///
<param name="pixelX">
Output parameter receiving the X coordinate in pixels.
</param>
///
<param name="pixelY">
Output parameter receiving the Y coordinate in pixels.
</param>
public
static
void
LatLongToPixelXY(
double
latitude,
double
longitude,
int
levelOfDetail,
out
int
pixelX,
out
int
pixelY)
{
latitude
=
Clip(latitude, MinLatitude, MaxLatitude);
longitude
=
Clip(longitude, MinLongitude, MaxLongitude);
double
x
=
(longitude
+
180
)
/
360
;
double
sinLatitude
=
Math.Sin(latitude
*
Math.PI
/
180
);
double
y
=
0.5
-
Math.Log((
1
+
sinLatitude)
/
(
1
-
sinLatitude))
/
(
4
*
Math.PI);
uint
mapSize
=
MapSize(levelOfDetail);
pixelX
=
(
int
) Clip(x
*
mapSize
+
0.5
,
0
, mapSize
-
1
);
pixelY
=
(
int
) Clip(y
*
mapSize
+
0.5
,
0
, mapSize
-
1
);
}
///
<summary>
///
Converts a pixel from pixel XY coordinates at a specified level of detail
///
into latitude/longitude WGS-84 coordinates (in degrees).
///
</summary>
///
<param name="pixelX">
X coordinate of the point, in pixels.
</param>
///
<param name="pixelY">
Y coordinates of the point, in pixels.
</param>
///
<param name="levelOfDetail">
Level of detail, from 1 (lowest detail)
///
to 23 (highest detail).
</param>
///
<param name="latitude">
Output parameter receiving the latitude in degrees.
</param>
///
<param name="longitude">
Output parameter receiving the longitude in degrees.
</param>
public
static
void
PixelXYToLatLong(
int
pixelX,
int
pixelY,
int
levelOfDetail,
out
double
latitude,
out
double
longitude)
{
double
mapSize
=
MapSize(levelOfDetail);
double
x
=
(Clip(pixelX,
0
, mapSize
-
1
)
/
mapSize)
-
0.5
;
double
y
=
0.5
-
(Clip(pixelY,
0
, mapSize
-
1
)
/
mapSize);
latitude
=
90
-
360
*
Math.Atan(Math.Exp(
-
y
*
2
*
Math.PI))
/
Math.PI;
longitude
=
360
*
x;
}
///
<summary>
///
Converts pixel XY coordinates into tile XY coordinates of the tile containing
///
the specified pixel.
///
</summary>
///
<param name="pixelX">
Pixel X coordinate.
</param>
///
<param name="pixelY">
Pixel Y coordinate.
</param>
///
<param name="tileX">
Output parameter receiving the tile X coordinate.
</param>
///
<param name="tileY">
Output parameter receiving the tile Y coordinate.
</param>
public
static
void
PixelXYToTileXY(
int
pixelX,
int
pixelY,
out
int
tileX,
out
int
tileY)
{
tileX
=
pixelX
/
256
;
tileY
=
pixelY
/
256
;
}
///
<summary>
///
Converts tile XY coordinates into pixel XY coordinates of the upper-left pixel
///
of the specified tile.
///
</summary>
///
<param name="tileX">
Tile X coordinate.
</param>
///
<param name="tileY">
Tile Y coordinate.
</param>
///
<param name="pixelX">
Output parameter receiving the pixel X coordinate.
</param>
///
<param name="pixelY">
Output parameter receiving the pixel Y coordinate.
</param>
public
static
void
TileXYToPixelXY(
int
tileX,
int
tileY,
out
int
pixelX,
out
int
pixelY)
{
pixelX
=
tileX
*
256
;
pixelY
=
tileY
*
256
;
}
///
<summary>
///
Converts tile XY coordinates into a QuadKey at a specified level of detail.
///
</summary>
///
<param name="tileX">
Tile X coordinate.
</param>
///
<param name="tileY">
Tile Y coordinate.
</param>
///
<param name="levelOfDetail">
Level of detail, from 1 (lowest detail)
///
to 23 (highest detail).
</param>
///
<returns>
A string containing the QuadKey.
</returns>
public
static
string
TileXYToQuadKey(
int
tileX,
int
tileY,
int
levelOfDetail)
{
StringBuilder quadKey
=
new
StringBuilder();
for
(
int
i
=
levelOfDetail; i
>
0
; i
--
)
{
char
digit
=
'
0
'
;
int
mask
=
1
<<
(i
-
1
);
if
((tileX
&
mask)
!=
0
)
{
digit
++
;
}
if
((tileY
&
mask)
!=
0
)
{
digit
++
;
digit
++
;
}
quadKey.Append(digit);
}
return
quadKey.ToString();
}
///
<summary>
///
Converts a QuadKey into tile XY coordinates.
///
</summary>
///
<param name="quadKey">
QuadKey of the tile.
</param>
///
<param name="tileX">
Output parameter receiving the tile X coordinate.
</param>
///
<param name="tileY">
Output parameter receiving the tile Y coordinate.
</param>
///
<param name="levelOfDetail">
Output parameter receiving the level of detail.
</param>
public
static
void
QuadKeyToTileXY(
string
quadKey,
out
int
tileX,
out
int
tileY,
out
int
levelOfDetail)
{
tileX
=
tileY
=
0
;
levelOfDetail
=
quadKey.Length;
for
(
int
i
=
levelOfDetail; i
>
0
; i
--
)
{
int
mask
=
1
<<
(i
-
1
);
switch
(quadKey[levelOfDetail
-
i])
{
case
'
0
'
:
break
;
case
'
1
'
:
tileX
|=
mask;
break
;
case
'
2
'
:
tileY
|=
mask;
break
;
case
'
3
'
:
tileX
|=
mask;
tileY
|=
mask;
break
;
default
:
throw
new
ArgumentException(
"
Invalid QuadKey digit sequence.
"
);
}
}
}
}
}
注:本文中内容来源于互联网整理而成,如涉及到任何侵权等行为请联系本人。
推荐资源:
《Bing Maps Tile System》
转载自http://www.cnblogs.com/beniao/archive/2010/04/18/1714544.html